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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク sociable number ： ウィキペディア英語版 sociable number Sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a set of sociable numbers, each number is the sum of the proper factors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point. The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3. It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound. An example with period 4: :The sum of the proper divisors of $1264460$ ($=2^2\backslash cdot5\backslash cdot17\backslash cdot3719$) is: ::1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860 :The sum of the proper divisors of $1547860$ ($=2^2\backslash cdot5\backslash cdot193\backslash cdot401$) is: ::1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636 :The sum of the proper divisors of $1727636$ ($=2^2\backslash cdot521\backslash cdot829$) is: ::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184 :The sum of the proper divisors of $1305184$ ($=2^5\backslash cdot40787$) is: ::1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460. The following categorizes all known sociable numbers as of November 2015 by the length of the corresponding aliquot sequence: == Searching for sociable numbers == The aliquot sequence can be represented as a directed graph, $G\_$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$. Cycles in $G\_$ represent sociable numbers within the interval $()$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「sociable number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.